Implicit copulas
have a long history with key developments spread across multiple fields,
including actuarial studies, econometrics, operations research,
probability and statistics.
Yet while there are many excellent existing monographs and surveys on copulas and
copula models (see [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]} and [10]} for prominent examples)
there does not appear
to be a dedicated survey or overview on this important class of copulas.
This paper aims to fill this gap and provides an overview that stresses common
features of the implicit copula family, likelihood-based estimation, and the usefulness
of implicit copulas in statistical
modeling. Particular focus is given to recent developments on
implicit copula processes for regression
and time series data, along with Bayesian inference that extends the
earlier overview by [11]} to these copula processes.
